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21
Exact Enumeration Of 1342Avoiding Permutations A Close Link With Labeled Trees And Planar Maps
, 1997
"... Solving the first nonmonotonic, longerthanthree instance of a classic enumeration problem, we obtain the generating function H(x) of all 1342avoiding permutations of length n as well as an exact formula for their number Sn (1342). While achieving this, we bijectively prove that the number of inde ..."
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Cited by 84 (7 self)
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Solving the first nonmonotonic, longerthanthree instance of a classic enumeration problem, we obtain the generating function H(x) of all 1342avoiding permutations of length n as well as an exact formula for their number Sn (1342). While achieving this, we bijectively prove that the number of indecomposable 1342avoiding permutations of length n equals that of labeled plane trees of a certain type on n vertices recently enumerated by Cori, Jacquard and Schaeffer, which is in turn known to be equal to the number of rooted bicubic maps enumerated by Tutte in 1963. Moreover, H(x) turns out to be algebraic, proving the first nonmonotonic, longerthanthree instance of a conjecture of Zeilberger and Noonan. We also prove that n p Sn (1342) converges to 8, so in particular, limn!1 (Sn (1342)=Sn (1234)) = 0.
Spiders for rank 2 Lie algebras
 Commun. Math. Phys
, 1996
"... Abstract. A spider is an axiomatization of the representation theory of a group, quantum group, Lie algebra, or other group or grouplike object. It is also known as a spherical category, or a strict, monoidal category with a few extra properties, or by several other names. A recently useful point o ..."
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Cited by 62 (1 self)
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Abstract. A spider is an axiomatization of the representation theory of a group, quantum group, Lie algebra, or other group or grouplike object. It is also known as a spherical category, or a strict, monoidal category with a few extra properties, or by several other names. A recently useful point of view, developed by other authors, of the representation theory of sl(2) has been to present it as a spider by generators and relations. That is, one has an algebraic spider, defined by invariants of linear representations, and one identifies it as isomorphic to a combinatorial spider, given by generators and relations. We generalize this approach to the rank 2 simple Lie algebras, namely A2, B2, and G2. Our combinatorial rank 2 spiders yield bases for invariant spaces which are probably related to Lusztig’s canonical bases, and they are useful for computing quantities such as generalized 6jsymbols and quantum link invariants. Their definition originates in definitions of the rank 2 quantum link invariants that were discovered independently by the author and Francois Jaeger. 1.
Cyclic SelfDual Codes
, 1983
"... It is shown that if the automorphism group of a binary selfdual code satisfies a certain condition then the code contains words of weight congruent to 2 modulo 4. In particular, no cyclic binary selfdual code can have all its weights divisible by 4. The number of cyclic binary selfdual codes of l ..."
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Cited by 43 (5 self)
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It is shown that if the automorphism group of a binary selfdual code satisfies a certain condition then the code contains words of weight congruent to 2 modulo 4. In particular, no cyclic binary selfdual code can have all its weights divisible by 4. The number of cyclic binary selfdual codes of length n is determined, and the shortest nontrivial code in this class is shown to have length 14. 1.
Complexes of Directed Trees
 J. Combin. Theory Ser. A
, 1998
"... To every directed graph G one can associate a complex \Delta(G) consisting of directed subforests. This construction, suggested to us by R. Stanley, is especially important in the case of a complete double directed graph Gn , where it leads to studying some interesting representations of the symm ..."
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Cited by 29 (5 self)
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To every directed graph G one can associate a complex \Delta(G) consisting of directed subforests. This construction, suggested to us by R. Stanley, is especially important in the case of a complete double directed graph Gn , where it leads to studying some interesting representations of the symmetric group and corresponds (via StanleyReisner correspondence) to an interesting quotient ring.
The quantum G2 link invariant
 International Journal of Math
"... We derive an inductive, combinatorial definition of a polynomialvalued regular isotopy invariant of links and tangled graphs. We show that the invariant equals the ReshetikhinTuraev invariant corresponding to the exceptional simple Lie algebra G2. It is therefore related to G2 in the same way that ..."
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Cited by 15 (1 self)
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We derive an inductive, combinatorial definition of a polynomialvalued regular isotopy invariant of links and tangled graphs. We show that the invariant equals the ReshetikhinTuraev invariant corresponding to the exceptional simple Lie algebra G2. It is therefore related to G2 in the same way that the HOMFLY polynomial is related to An and the Kauffman polynomial is related to Bn, Cn, and Dn. We give parallel constructions for the other rank 2 Lie algebras and present some combinatorial conjectures motivated by the new inductive definitions. This paper is divided into two parts. In the first part we derive from first principles some variants of the link invariant known as the Jones polynomial. In the second part we show that these invariants are the same known invariants constructed using rank 2 Lie algebras, and we discuss some of their properties. 1 Invariants of links and graphs The simplest known definition of the Jones polynomial is the Kauffman bracket [8], which in this paper will be denoted by 〈·〉A1 and will be called the A1 bracket. The A1 bracket is given by the following recursive rules:  〉 A1 = −(q 1/2 +q1/2)  〉 A1  〉 A1 = −q 1/4  〉 A1 − q1/4  〉 A1 The goal of this part of the paper is to derive definitions of the following three variants of the Jones polynomial: Theorem 1.1. There is an invariant for regular isotopy of projections of links and tangled trivalent graphs called 〈·〉G2 which is given by the following recursive rules:
The pathcycle symmetric function of a digraph
 Adv. Math
, 1996
"... Recently, Stanley [21] has defined a symmetric function generalization of the chromatic polynomial of a graph. Independently, Chung and Graham [4] have defined a digraph polynomial called the cover polynomial which is closely related to the chromatic polynomial of a graph (in fact, as ..."
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Cited by 14 (4 self)
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Recently, Stanley [21] has defined a symmetric function generalization of the chromatic polynomial of a graph. Independently, Chung and Graham [4] have defined a digraph polynomial called the cover polynomial which is closely related to the chromatic polynomial of a graph (in fact, as
A complementation theorem for perfect matchings of graphs having a cellular completion
 J. Combin. Theory Ser. A
, 1998
"... Abstract. A cellular graph is a graph whose edges can be partitioned into 4cycles (called cells) so that each vertex is contained in at most two cells. We present a “Complementation Theorem ” for the number of matchings of certain subgraphs of cellular graphs. This generalizes the main result of [2 ..."
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Cited by 11 (3 self)
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Abstract. A cellular graph is a graph whose edges can be partitioned into 4cycles (called cells) so that each vertex is contained in at most two cells. We present a “Complementation Theorem ” for the number of matchings of certain subgraphs of cellular graphs. This generalizes the main result of [2]. As applications of the Complementation Theorem we obtain a new proof of Stanley’s multivariate version of the Aztec diamond theorem, a weighted generalization of a result of Knuth [7] concerning spanning trees of Aztec diamond graphs, a combinatorial proof of Yang’s enumeration [17] of matchings of fortress graphs and direct proofs for certain identities of Jockusch and Propp [6]. 1.
A HopfAlgebra Approach To Inner Plethysm
"... We use the Hopf algebra structure of the algebra of symmetric functions to study the Adams operators of the complex representation rings of symmetric groups, and we give new proofs of all of Littlewood's formulas for inner plethysm. We also study the Adams operations for orthogonal and symplectic gr ..."
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Cited by 9 (6 self)
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We use the Hopf algebra structure of the algebra of symmetric functions to study the Adams operators of the complex representation rings of symmetric groups, and we give new proofs of all of Littlewood's formulas for inner plethysm. We also study the Adams operations for orthogonal and symplectic group characters. 1 Background Our notations for symmetric functions (of an infinite, countable set of indeterminates X = fx 1 ; x 2 ; : : :g) will be the same as in [Mcd], except for the following minor changes. We denote by Sym(X) := n Sym (X) the Zalgebra of symmetric functions over X , graded by total degree. The standard scalar product on Sym(X) (for which the Schur functions form an orthonormal basis) is denoted by (\Delta ; \Delta) rather than by h\Delta ; \Deltai, the symbol h\Deltai being reserved for an other use. Also, for a symmetric function F , we denote by D F instead of D(F ) the adjoint of the linear operator G 7! FG. The generating series for complete and elementary functions are oe z (X) = h n (X) = (1 \Gamma zx i ) z (X) = e n (X) = (1 + zx i ) : A partition can be described by the sequence of its parts, arranged in decreasing order or by the multiplicities of the parts. If ff = (ff 1 ; ff 2 ; : : :) = (1 : : :), then we call the sequence a := (a 1 ; a 2 ; : : :) the cycle type of ff.
Fast FloatingPoint Processing in Common Lisp
 ACM Trans. on Math. Software
, 1995
"... this paper we explore an approach which enables all of the problems listed above to be solved at a single stroke: use Lisp as the source language for the numeric and graphical code! This is not a new idea  it was tried at MIT and UCB in the 1970's. While these experiments were modestly successful ..."
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Cited by 5 (1 self)
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this paper we explore an approach which enables all of the problems listed above to be solved at a single stroke: use Lisp as the source language for the numeric and graphical code! This is not a new idea  it was tried at MIT and UCB in the 1970's. While these experiments were modestly successful, the particular systems are obsolete. Fortunately, some of those ideas used in Maclisp [37], NIL [38] and Franz Lisp [20] were incorporated in the subsequent standardization of Common Lisp (CL) [35]. In this new setting it is appropriate to reexamine the theoretical and practical implications of writing numeric code in Lisp. The popular conceptions of Lisp's inefficiency for numerics have been based on rumor, supposition, and experience with early and (in fact) inefficient implementations. It is certainly possible to continue to write inefficient programs: As one example of the results of deemphasizing numerics in the design, consider the situation of the basic arithmetic operators. The definitions of these functions require that they are generic, (e.g. "+" must be able to add any combination of several precisions of floats, arbitraryprecision integers, rational numbers, and complexes), The very simple way of implementing this arithmetic  by subroutine calls  is also very inefficient. Even with appropriate declarations to enable more specific treatment of numeric types, compilers are free to ignore declarations and such implementations naturally do not accommodate the needs of intensive numbercrunching. (See the appendix for further discussion of declarations). Be this as it may, the situation with respect to Lisp has changed for the better in recent years. With the advent of ANSI standard Common Lisp, several active vendors of implementations and one active universi...
Nonholonomicity of sequences defined via elementary functions
 Annals of Combinatorics
"... We present a new method for proving nonholonomicity of sequences, which is based on results about the number of zeros of elementary and of analytic functions. Our approach is applicable to sequences that are defined as the values of an elementary function at positive integral arguments. We generali ..."
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Cited by 5 (3 self)
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We present a new method for proving nonholonomicity of sequences, which is based on results about the number of zeros of elementary and of analytic functions. Our approach is applicable to sequences that are defined as the values of an elementary function at positive integral arguments. We generalize several recent results; e.g., nonholonomicity of the logarithmic sequence is extended to rational functions involving log n. Moreover, we show that the sequence that arises from evaluating the Riemann zeta function at odd integers is not holonomic.